The beginnings of the probability calculus, with special reference to the works of De Moivre and Simpson

This study is concerned with aspects of the early history of the Probability Calculus up to the time of its formalisation and its application to the mid-eighteenth century theory of errors. Commencing with an analysis of early enumeration. of the outcomes of three dice recorded in the poem De Vetula, a brief history of the introduction of cards to Europe followed and a link between religion and games of chance is suggested. The fruitful endeavours of Cardano are then examined as he tried to advance beyond the simple enumeration techniques for repeated events. Contributions by Galileo, Fermat and Pascal are discussed. Huygens' independently introduced concept of "expectation" was popular with English mathematicians at the end of the seventeenth century. The use of the binomial theorem for a large number of outcomes was recognised by Jacques Bernoulli as being appropriate, and by his mathematical ingenuity he reached the Law of Large Numbers. With the introduction of the definition of probability as a ratio and the powerful mathematical methods of infinite series, binomial theorem and infinitesimal calculus, rapid progress was made by Arbuthnot, Montmort and De Moivre which resulted in the approximation n.?2?n.(n/e)n, and in the case of the binomial expansion, the ratio of a term distant ? from the middle term to the sum of the terms was obtained as ?2/(?n). e-2 ? 2/n, approximately. Simpson extended De Moivre's discoveries to apply the first theoretical probability distribution to a number of hypothetical observations and thereby illustrated the advantage of using the mean of numerous observations